Concrete mathematics : a foundation for computer science

4.5 RELATIVE PRIMALITY 117
and then we’ll get 8, 16, and so on. The entire array can be regarded as an
/guess l/O is infinite binary tree structure whose top levels look like this:
infinity, “in lowest
terms.” n 1
Conserve parody.
Each fraction is *, where F is the nearest ancestor above and to the left,
and $ is the nearest ancestor above and to the right. (An “ancestor” is a
fraction that’s reachable by following the branches upward.) Many patterns
can be observed in this tree.
Why does this construction work? Why, for example, does each mediant
fraction (mt m’)/(n +n’) turn out to be in lowest terms when it appears in
this tree? (If m, m’, n, and n’ were all odd, we’d get even/even; somehow the
construction guarantees that fractions with odd numerators and denominators
never appear next to each other.) And why do all possible fractions m/n occur
exactly once? Why can’t a particular fraction occur twice, or not at all?
All of these questions have amazingly simple answers, based on the following
fundamental fact: If m/n and m//n’ are consecutive fractions at any
stage of the construction, we have
m’n-mn’ = 1. (4.31)
This relation is true initially (1 . 1 - 0.0 = 1); and when we insert a new
mediant (m + m’)/(n + n’), the new cases that need to be checked are
(m+m’)n-m(n+n’) = 1;
m’(n + n’) - (m + m’)n’ = 1 .
Both of these equations are equivalent to the original condition (4.31) that
they replace. Therefore (4.31) is invariant at all stages of the construction.
Furthermore, if m/n < m’/n’ and if all values are nonnegative, it’s easy
to verify that
m/n < (m-t m’)/(n+n’) < m’/n’.